Lemma 4.2 (See [1]}, Lemma 1.5 and a historical remark preceding its formulation.) 

Let \((X, \mathcal {B}, \mu )\)  be a probability space. Let \(f_n(x)\)  be a sequence of non-negative measurable functions on \(X\) . Let \(u_n, v_n\)  be sequences of real numbers such that
\( 0 \le u_n \le v_n.\) 
